# Flow to Rare Events: An Application of Normalizing Flow in Temporal Importance Sampling for Automated Vehicle Validation

arXiv: https://arxiv.org/abs/2407.07320 · HTML: https://mlsanity.com/paper/2407.07320/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 33 references that are in the map.
- First posted on arXiv 10 Jul 2024.

## Builds on (most cited first, 17 of 17)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [The highD Dataset: A Drone Dataset of Naturalistic Vehicle Trajectories on German Highways for Validation of Highly Automated Driving Systems](https://mlsanity.com/paper/1810.05642/) (2018)
- ["Curse of rarity" for autonomous vehicles](https://mlsanity.com/paper/2207.02749/) (2022)
- [Accelerated Evaluation of Automated Vehicles Safety in Lane Change Scenarios Based on Importance Sampling Techniques](https://mlsanity.com/paper/1605.04965/) (2016)
- [Ontology based Scene Creation for the Development of Automated Vehicles](https://mlsanity.com/paper/1704.01006/) (2017)
- [Formal Scenario-Based Testing of Autonomous Vehicles: From Simulation to the Real World](https://mlsanity.com/paper/2003.07739/) (2020)
- [Testing Scenario Library Generation for Connected and Automated Vehicles, Part I: Methodology](https://mlsanity.com/paper/1905.03419/) (2019)
- [Waymo Public Road Safety Performance Data](https://mlsanity.com/paper/2011.00038/) (2020)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Temporal Normalizing Flows](https://mlsanity.com/paper/1912.09092/) (2019)
- [An Accelerated Testing Approach for Automated Vehicles with Background Traffic Described by Joint Distributions](https://mlsanity.com/paper/1707.04896/) (2017)
- [Probabilistic Metamodels for an Efficient Characterization of Complex Driving Scenarios](https://mlsanity.com/paper/2110.02892/) (2021)
- [Test Against High-Dimensional Uncertainties: Accelerated Evaluation of Autonomous Vehicles with Deep Importance Sampling](https://mlsanity.com/paper/2204.02351/) (2022)

## Built on by (most cited first, 1 of 1)

- [Jeffreys Flow: Robust Boltzmann Generators for Rare Event Sampling via Parallel Tempering Distillation](https://mlsanity.com/paper/2604.05303/) (2026)

## Abstract (arXiv)

Automated Vehicle (AV) validation based on simulated testing requires unbiased evaluation and high efficiency. One effective solution is to increase the exposure to risky rare events while reweighting the probability measure. However, characterizing the distribution of risky events is particularly challenging due to the paucity of samples and the temporality of continuous scenario variables. To solve it, we devise a method to represent, generate, and reweight the distribution of risky rare events. We decompose the temporal evolution of continuous variables into distribution components based on conditional probability. By introducing the Risk Indicator Function, the distribution of risky rare events is theoretically precipitated out of naturalistic driving distribution. This targeted distribution is practically generated via Normalizing Flow, which achieves exact and tractable probability evaluation of intricate distribution. The rare event distribution is then demonstrated as the advantageous Importance Sampling distribution. We also promote the technique of temporal Importance Sampling. The combined method, named as TrimFlow, is executed to estimate the collision rate of Car-following scenarios as a tentative practice. The results showed that sampling background vehicle maneuvers from rare event distribution could evolve testing scenarios to hazardous states. TrimFlow reduced 86.1% of tests compared to generating testing scenarios according to their exposure in the naturalistic driving environment. In addition, the TrimFlow method is not limited to one specific type of functional scenario.